Self-adapting feedforward control tuning for motion system, and lithographic apparatus provided with such a motion system

ABSTRACT

A motion control system is presented. In an embodiment of the invention, the motion control system compares a motion profile setpoint generator configured to generate a set of profile setpoint signals including a position profile setpoint signal and additional profile setpoint signals. The feedforward controller generates a feedforward signal by summing the additional profile setpoint signals, the additional profile setpoint signals being multiplied by a respective coefficient. The motion control system is configured to (a) select an initial setting of the respective coefficients, (b) cause the motion profile setpoint generator to generate a set of profile setpoint signals so as to execute a motion profile, (c) measure an error between the position profile and an output position signal, and (d) update the respective coefficients by adding a product of the error and a respective learning gain thereto. In an implementation, standard motion controllers can be used.

FIELD

The present invention relates to a lithographic apparatus and moreparticularly to a lithographic apparatus provided with a motion system.

BACKGROUND

In many consumer products and manufacturing machines, high-precisionmotion systems may be embedded. Examples of consumer products withhigh-precision motion systems are hard disk drives, optical drives, andtape drives. Examples of manufacturing machines with high-precisionmotion systems are scanning lithography stages, and pick and placerobots.

A typical controller for a high-precision motion system consists of afeedback controller and a feedforward controller. The feedforwardcontroller uses the knowledge on the positioning setpoint to generate afeedforward command in such a way that the open loop response (with thefeedback controller inactive) of the motion system resembles thesetpoint as closely as possible. Different types of feedforward controlcan be distinguished. In the next paragraphs, two methods are discussedin more detail: Low-order Feedforward Control and Iterative LearningControl.

Low-order feedforward control (LFC) is well known and widely spread inindustry. A low-order feedforward controller often includes differentparts. A first part may be related to the setpoint acceleration, togenerate the part of the command that is inertia-related. A second partmay be related to the setpoint velocity, to generate the part of thecommand that is damping-related or friction-related. In the mostadvanced implementations, a third part may be related to the setpointsnap (identical to the derivative of jerk), to generate the part of thecommand that is related to the elastic deformation of the motion system.Advantages of LFC are its simplicity, its ease of implementation, andits flexibility against variations in the motion profile to be realized.A disadvantage of LFC is the required effort and time spent in manuallytuning the coefficients of the different feedforward parts. Anotherdisadvantage of LFC is the fact that the feedforward controller may notadapt to variations, in particular slow variations, in the motionsystem, in which case the performance will degrade.

Iterative Learning Control (ILC) is a methodology, which updates thecontrol signal for a repeating task iteratively in such a way that thedifference between the desired and the actual behavior of thesystem-to-be-controlled vanishes. ILC has been successfully applied inseveral applications. However, the main strength of ILC is also its mainweakness: for repeating tasks, excellent feedforward signals can bedesigned, whereas these feedforward signals are useless for motionprofiles with different characteristics, such as maximum displacement,velocity, acceleration. Thus, ILC may be inflexible, or, in the bestcase, only partially flexible for different motion profiles. Anotherdisadvantage of ILC is that it usually needs rather extensive on-linecomputing and the storage and update of lengthy feedforward tables. Anadvantage of ILC is its ability to adapt to slow variations in themotion system, for example to changes in the inertia of the system or inmotor constants.

An example of an apparatus including several high-precision motionsystems is a lithographic apparatus, which is described hereinafter.

A lithographic apparatus is a machine that applies a desired patternonto a substrate, usually onto a target portion of the substrate. Alithographic apparatus can be used, for example, in the manufacture ofintegrated circuits (ICs). In that instance, a patterning device, whichis alternatively referred to as a mask or a reticle, may be used togenerate a circuit pattern to be formed on an individual layer of theIC. This pattern can be transferred onto a target portion (e.g.including part of, one, or several dies) on a substrate (e.g. a siliconwafer). Transfer of the pattern is typically via imaging onto a layer ofradiation-sensitive material (resist) provided on the substrate. Ingeneral, a single substrate will contain a network of adjacent targetportions that are successively patterned. Conventional lithographicapparatus include so-called steppers, in which each target portion isirradiated by exposing an entire pattern onto the target portion atonce, and so-called scanners, in which each target portion is irradiatedby scanning the pattern through a radiation beam in a given direction(the “scanning”-direction) while synchronously scanning the substrateparallel or anti-parallel to this direction. It is also possible totransfer the pattern from the patterning device to the substrate byimprinting the pattern onto the substrate.

In a lithographic apparatus, high-precision motion systems may be foundfor performing the stepping and scanning operations.

SUMMARY

Embodiments of the invention include a motion control system enablingthe automation of feedforward tuning.

According to an embodiment of the invention, a motion control systemincludes a motion profile setpoint generator for generating a set ofprofile setpoint signals including a position profile setpoint signaland additional profile setpoint signals; a feedforward controller forgenerating a feedforward signal by summing the additional profilesetpoint signals, the additional profile setpoint signals beingmultiplied by a respective coefficient; the motion control system beingconfigured to: (a) select an initial setting of the respectivecoefficients; (b) cause the motion profile setpoint generator togenerate a set of profile setpoint signals so as to execute a motionprofile; (c) measure an error between the position profile and an outputposition signal; (d) update the respective coefficients by adding aproduct of the error and a respective learning gain thereto.

According to an embodiment of the invention, a method for feedforwardmotion control includes generating a set of profile setpoint signalsincluding a position profile setpoint signal and additional profilesetpoint signals; generating a feedforward signal by summing theadditional profile setpoint signals, the additional profile setpointsignals being multiplied by a respective coefficient; and determiningthe coefficients by: (a) selecting an initial setting of the respectivecoefficients; (b) generating a set of profile setpoint signals so as toexecute a motion profile; (c) measuring an error between the positionprofile and an output position signal; (d) updating the respectivecoefficients by adding a product of the error and a respective learninggain thereto.

According to an embodiment of the invention, a lithographic apparatus isprovided, the lithographic apparatus being arranged to transfer apattern from a patterning device onto a substrate, wherein at least oneof the patterning device and the substrate is positioned with apositioner which is controlled by a motion control system, the motioncontrol system including a motion profile setpoint generator forgenerating a set of profile setpoint signals including a positionprofile setpoint signal and additional profile setpoint signals; afeedforward controller for generating a feedforward signal by summingthe additional profile setpoint signals, the additional profile setpointsignals being multiplied by a respective coefficient; the motion controlsystem being configured to: (a) select an initial setting of therespective coefficients; (b) cause the motion profile setpoint generatorto generate a set of profile setpoint signals so as to execute a motionprofile; (c) measure an error between the position profile and an outputposition signal; (d) update the respective coefficients by adding aproduct of the error and a respective learning gain thereto.

The invention provides for updating the coefficients of a low-orderfeedforward controller iteratively such that the difference between thedesired and the actual behavior of the motion system is compensated.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described, by way of exampleonly, with reference to the accompanying schematic drawings in whichcorresponding reference symbols indicate corresponding parts, and inwhich:

FIG. 1 depicts a lithographic apparatus according to an embodiment ofthe invention;

FIG. 2 shows a block diagram of a motion control system according to anembodiment of the invention;

FIG. 3 shows three plots of an acceleration signal as a function oftime;

FIG. 4 shows a frequency response function (magnitude and phase,respectively, as a function of frequency) of the motion control systemin accordance with an embodiment of the invention, as well as a modelthereof;

FIG. 5 shows timing charts of a setpoint snap, acceleration, andposition, respectively, as a function of time, for the system of FIG. 4;

FIG. 6 shows a timing chart of a motion control error, with (solid line)and without (broken line) snap tuning, in the absence of a measurementdelay;

FIG. 7 illustrates a convergence over several iterations of anacceleration coefficient kfa and a snap coefficient kfs, respectively,in the absence of a measurement delay;

FIG. 8 shows a timing chart of a motion control error, with (solid line)and without (broken line) delay compensation tuning, in the presence ofa measurement delay; and

FIG. 9 illustrates a convergence over several iterations of accelerationcoefficients ratio kfa₀/kfa₁ and snap coefficients ratio kfs₀/kfs₁,respectively, in the presence of a measurement delay.

DETAILED DESCRIPTION

FIG. 1 schematically depicts a lithographic apparatus according to oneembodiment of the invention. The apparatus includes an illuminationsystem (illuminator) IL configured to condition a radiation beam B (e.g.UV radiation or any other type of suitable radiation) and a supportstructure (e.g. a mask table) MT constructed to support a patterningdevice (e.g. a mask) MA and connected to a first positioner PMconfigured to accurately position the patterning device in accordancewith certain parameters. The apparatus also includes a substrate table(e.g. a wafer table) WT constructed to hold a substrate (e.g. aresist-coated wafer) W and connected to a second positioner PWconfigured to accurately position the substrate in accordance withcertain parameters; and a projection system (e.g. a refractiveprojection lens system) PS configured to project a pattern imparted tothe radiation beam B by patterning device MA onto a target portion C(e.g. including one or more dies) of the substrate W.

The first positioner PM, the second positioner PW, and any otherpositioner included in the apparatus each may include a motion controlsystem according to the invention. Such a motion control system will beexplained in more detail below.

The illumination system may include various types of optical components,such as refractive, reflective, magnetic, electromagnetic, electrostaticor other types of optical components, or any combination thereof, fordirecting, shaping, or controlling radiation.

The support structure supports, i.e. bears the weight of, the patterningdevice. It holds the patterning device in a manner that depends on theorientation of the patterning device, the design of the lithographicapparatus, and other conditions, such as for example whether or not thepatterning device is held in a vacuum environment. The support structurecan use mechanical, vacuum, electrostatic or other clamping techniquesto hold the patterning device. The support structure may be a frame or atable, for example, which may be fixed or movable as required. Thesupport structure may ensure that the patterning device is at a desiredposition, for example with respect to the projection system. Any use ofthe terms “reticle” or “mask” herein may be considered synonymous withthe more general term “patterning device.”

The term “patterning device” used herein should be broadly interpretedas referring to any device that can be used to impart a radiation beamwith a pattern in its cross-section such as to create a pattern in atarget portion of the substrate. It should be noted that the patternimparted to the radiation beam may not exactly correspond to the desiredpattern in the target portion of the substrate, for example if thepattern includes phase-shifting features or so called assist features.Generally, the pattern imparted to the radiation beam will correspond toa particular functional layer in a device being created in the targetportion, such as an integrated circuit.

The patterning device may be transmissive or reflective. Examples ofpatterning devices include masks, programmable mirror arrays, andprogrammable LCD panels. Masks are well known in lithography, andinclude mask types such as binary, alternating phase-shift, andattenuated phase-shift, as well as various hybrid mask types. An exampleof a programmable mirror array employs a matrix arrangement of smallmirrors, each of which can be individually tilted so as to reflect anincoming radiation beam in different directions. The tilted mirrorsimpart a pattern in a radiation beam which is reflected by the mirrormatrix.

The term “projection system” used herein should be broadly interpretedas encompassing any type of projection system, including refractive,reflective, catadioptric, magnetic, electromagnetic and electrostaticoptical systems, or any combination thereof, as appropriate for theexposure radiation being used, or for other factors such as the use ofan immersion liquid or the use of a vacuum. Any use of the term“projection lens” herein may be considered as synonymous with the moregeneral term “projection system”.

As here depicted, the apparatus is of a transmissive type (e.g.employing a transmissive mask). Alternatively, the apparatus may be of areflective type (e.g. employing a programmable mirror array of a type asreferred to above, or employing a reflective mask).

The lithographic apparatus may be of a type having two (dual stage) ormore substrate tables (and/or two or more mask tables). In such“multiple stage” machines the additional tables may be used in parallel,or preparatory steps may be carried out on one or more tables while oneor more other tables are being used for exposure.

The lithographic apparatus may also be of a type wherein at least aportion of the substrate may be covered by a liquid having a relativelyhigh refractive index, e.g. water, so as to fill a space between theprojection system and the substrate. An immersion liquid may also beapplied to other spaces in the lithographic apparatus, for example,between the mask and the projection system. Immersion techniques arewell known in the art for increasing the numerical aperture ofprojection systems. The term “immersion” as used herein does not meanthat a structure, such as a substrate, must be submerged in liquid, butrather only means that liquid is located between the projection systemand the substrate during exposure.

Referring to FIG. 1, the illuminator IL receives a radiation beam from aradiation source SO. The source and the lithographic apparatus may beseparate entities, for example when the source is an excimer laser. Insuch cases, the source is not considered to form part of thelithographic apparatus and the radiation beam is passed from the sourceSO to the illuminator IL with the aid of a beam delivery system BDincluding, for example, suitable directing mirrors and/or a beamexpander. In other cases the source may be an integral part of thelithographic apparatus, for example when the source is a mercury lamp.The source SO and the illuminator IL, together with the beam deliverysystem BD if needed, may be referred to as a radiation system.

The illuminator IL may include an adjuster AD for adjusting the angularintensity distribution of the radiation beam. Generally, at least theouter and/or inner radial extent (commonly referred to as σ-outer andσ-inner, respectively) of the intensity distribution in a pupil plane ofthe illuminator can be adjusted. In addition, the illuminator IL mayinclude various other components, such as an integrator IN and acondenser CO. The illuminator may be used to condition the radiationbeam, to have a desired uniformity and intensity distribution in itscross-section.

The radiation beam B is incident on the patterning device (e.g., maskMA), which is held on the support structure (e.g., mask table MT), andis patterned by the patterning device. Having traversed the mask MA, theradiation beam B passes through the projection system PS, which focusesthe beam onto a target portion C of the substrate W. With the aid of thesecond positioner PW and position sensor IF (e.g. an interferometricdevice, linear encoder or capacitive sensor), the substrate table WT canbe moved accurately, e.g. so as to position different target portions Cin the path of the radiation beam B. Similarly, the first positioner PMand another position sensor (which is not explicitly depicted in FIG. 1)can be used to accurately position the mask MA with respect to the pathof the radiation beam B, e.g. after mechanical retrieval from a masklibrary, or during a scan. In general, movement of the mask table MT maybe realized with the aid of a long-stroke module (coarse positioning)and a short-stroke module (fine positioning), which form part of thefirst positioner PM. Similarly, movement of the substrate table WT maybe realized using a long-stroke module and a short-stroke module, whichform part of the second positioner PW. In the case of a stepper (asopposed to a scanner) the mask table MT may be connected to ashort-stroke actuator only, or may be fixed. Mask MA and substrate W maybe aligned using mask alignment marks M1, M2 and substrate alignmentmarks P1, P2. Although the substrate alignment marks as illustratedoccupy dedicated target portions, they may be located in spaces betweentarget portions (these are known as scribe-lane alignment marks).Similarly, in situations in which more than one die is provided on themask MA, the mask alignment marks may be located between the dies.

The depicted apparatus could be used in at least one of the followingmodes:

Step mode: the mask table MT and the substrate table WT are keptessentially stationary, while an entire pattern imparted to theradiation beam is projected onto a target portion C at once (i.e. asingle static exposure). The substrate table WT is then shifted in the Xand/or Y direction so that a different target portion C can be exposed.In step mode, the maximum size of the exposure field limits the size ofthe target portion C imaged in a single static exposure.

Scan mode: the mask table MT and the substrate table WT are scannedsynchronously while a pattern imparted to the radiation beam isprojected onto a target portion C (i.e. a single dynamic exposure). Thevelocity and direction of the substrate table WT relative to the masktable MT may be determined by the (de-)magnification and image reversalcharacteristics of the projection system PS. In scan mode, the maximumsize of the exposure field limits the width (in the non-scanningdirection) of the target portion in a single dynamic exposure, whereasthe length of the scanning motion determines the height (in the scanningdirection) of the target portion.

Another mode: the mask table MT is kept essentially stationary holding aprogrammable patterning device, and the substrate table WT is moved orscanned while a pattern imparted to the radiation beam is projected ontoa target portion C. In this mode, generally a pulsed radiation source isemployed and the programmable patterning device is updated as requiredafter each movement of the substrate table WT or in between successiveradiation pulses during a scan. This mode of operation can be readilyapplied to maskless lithography that utilizes programmable patterningdevice, such as a programmable mirror array of a type as referred toabove.

Combinations and/or variations on the above described modes of use orentirely different modes of use may also be employed.

In a motion control system according to an embodiment of the invention,coefficients of a low-order feedforward controller are updatediteratively in such a way that the difference between the desired andthe actual behavior of the system to be controlled ceases. The updatemechanism is based on theory underlying the update mechanism used in theso-called lifted ILC framework, the main difference being that ILCupdates the individual samples of a feedforward signal, whereasaccording to an embodiment of the invention the coefficients of afeedforward controller are updated. Hence, these coefficients may betuned, after which the resulting feedforward controller can be appliedto arbitrary motion profiles. Convergence proofs known from ILC theorycan be used to prove the convergence of the update mechanism. Thus, avery useful combination of the industrially accepted LFC method(acceleration, velocity) and the powerful ILC method is made.

FIG. 2 shows a block diagram of a motion system with feedback andfeedforward control. s is the snap of the setpoint (derivative of jerk),j the jerk, a the acceleration, v the velocity and p the position. e isthe servo error, pi the plant input, y the plant output. FF is the totalfeedforward signal, consisting of a part that is equal to the sum of thesnap times kfs, the jerk times kfj, the acceleration times kfa and thevelocity times kfv. In many industrial applications, the snap and thejerk are not accessible for feedforward control, but for the presentinvention the snap and the jerk are included to present a more generalsituation. It is observed, however, that in an even more generalizedsituation also other profile setpoint signals may be defined, which havea defined relationship with the position signal. As an example, dryCoulomb friction may be mentioned, having a constant feedforward signalvalue with a sign which is equal to the sign of the velocity.

Taking into consideration snap, jerk, acceleration and velocity, thefollowing iterative procedure for updating the feedforward coefficientsmay be used in an embodiment of the invention:

-   -   Start with an initial setting for kfs, kfj, kfa, and kfv,        denoted as kfs⁰, kfj⁰, kfa⁰, and kfv⁰, respectively. Set        iteration counter k=1.    -   Execute a motion profile, measure the error e^(k) ε        ^(N) ^(k) ^(×1), where N^(k)=number of samples in the motion        profile used in iteration k.    -   Update the coefficients of the feedforward controller according        to the following update law:

$\begin{matrix}{\begin{bmatrix}{kfs}^{k + 1} \\{kfj}^{k + 1} \\{kfa}^{k + 1} \\{kfv}^{k + 1}\end{bmatrix} = {\begin{bmatrix}{kfs}^{k} \\{kfj}^{k} \\{kfa}^{k} \\{kfv}^{k}\end{bmatrix} + {L^{k}{\mathbb{e}}^{k}}}} & (1)\end{matrix}$where L^(k) ε

^(4×N) ^(k) is the learning gain. Set k=k+1 and go back to the secondstep.

More generally, this can be written as:

$\begin{bmatrix}{k1}^{k + 1} \\{k2}^{k + 1} \\\vdots \\{kn}^{k + 1}\end{bmatrix} = {\begin{bmatrix}{k1}^{k} \\{k2}^{k} \\\vdots \\{kn}^{k}\end{bmatrix} + {L^{k}{\mathbb{e}}^{k}}}$where:

-   -   k1^(k) . . . kn^(k) are respective coefficients, in a k^(th)        iteration, of n setpoint profile signals,    -   k1^(k+1) . . . kn^(k+1) are respective coefficients, in a        (k+1)^(th) iteration, of n setpoint profile signals,    -   L^(k) ε        ^(n×N) ^(k) is the learning gain in a k^(th) iteration,    -   N^(k) is a number of samples in the motion profile used in a        k^(th) iteration, and    -   e^(k) is the error in a k^(th) iteration.

The iteration described above can be repeated continuously to adapt toslow changes in the system dynamics. If these are not expected, theiterative procedure can be stopped after a number of iterations. In thiscase, the method is used as a calibration.

One possible way out of several ways of determination of L^(k) isdescribed next. L^(k) has to be updated every time a change is made inthe motion profile. If the motion profile does not change at all or doesnot change during calibration, a fixed learning gain L^(k)=L can beused, which can be computed off-line.

Let FF^(k) ε

^(N) ^(k) and y^(k) ε

^(N) ^(k) be the feedforward signal and measured output signal,respectively, in trial k. Let PS^(k) ε

^(N) ^(k) ^(×N) ^(k) be the map between FF and y .PS^(k) is a Toeplitzmatrix with the upper-diagonal triangular matrix equal to zero and thefirst column given by the impulse response coefficients of the processsensitivity (P/(1+PK)). Below, an example is given of how PS^(k) wouldlook for a motion system with impulse response coefficients [4,3,2,1]and N^(k)=4:

${PS}^{k} = \begin{bmatrix}4 & 0 & 0 & 0 \\3 & 4 & 0 & 0 \\2 & 3 & 4 & 0 \\1 & 2 & 3 & 4\end{bmatrix}$

There are several ways to derive PS^(k). One is to measure the impulseresponse of the system. Another is to use a simple model of the processsensitivity from which the impulse response coefficients are computed.The latter approach is used in the example described later.

The relation between a change in the feedforward parameters kfs^(k),kfj^(k), kfa^(k), and kfv^(k) and the corresponding change of the servoerror in trial k Δe^(k) is then given by the following expression:

$\begin{matrix}{{\Delta\mathbb{e}}^{k} = {- {{{PS}^{k}\begin{bmatrix}s^{k} & j^{k} & a^{k} & v^{k}\end{bmatrix}}\begin{bmatrix}{\Delta\;{kfs}^{k}} \\{\Delta\;{kfj}^{k}} \\{\Delta\;{kfa}^{k}} \\{\Delta\;{kfv}^{k}}\end{bmatrix}}}} & (2)\end{matrix}$

In each trial an update of the feedforward coefficients is computed suchthat the sum of squares of the error signal is minimized. Minimizing(e^(k)+Δe^(k))^(T) (e^(k)+Δe^(k)), with Δe^(k) given by (2) leads to thefollowing update law:

$\begin{matrix}{\begin{bmatrix}{\Delta\;{kfs}^{k}} \\{\Delta\;{kfj}^{k}} \\{\Delta\;{kfa}^{k}} \\{\Delta\;{kfv}^{k}}\end{bmatrix} = {{\left( {\begin{bmatrix}s^{k} & j^{k} & a^{k} & v^{k}\end{bmatrix}^{T}{PS}^{k^{T}}{{PS}^{k}\begin{bmatrix}s^{k} & j^{k} & a^{k} & v^{k}\end{bmatrix}}} \right)^{- 1}\begin{bmatrix}s^{k} & j^{k} & a^{k} & v^{k}\end{bmatrix}}^{T}{PS}^{k^{T}}{\mathbb{e}}^{k}}} & (3)\end{matrix}$

Based on (3), the learning gain in (1) is defined as follows:L ^(k)=α([s ^(k) j ^(k) a ^(k) v ^(k)]^(T) PS ^(k) ^(T) PS ^(k) [s ^(k)j ^(k) a ^(k) v ^(k)])⁻¹ [s ^(k) j ^(k) a ^(k) v ^(k)]^(T) PS ^(k) ^(T)  (4)where α is a learning factor which can be used to balance theconvergence speed against the insensitivity to non-repeating phenomenasuch as for example noise.

More generally, the learning gain L^(k) may be defined as:L ^(k)=α([m1^(k) m2^(k) . . . mn ^(k)]^(T) PS ^(k) ^(T) PS ^(k) [m1^(k)m2^(k) . . . mn ^(k)])⁻¹ [m1^(k) m2^(k) . . . mn ^(k)]^(T) PS ^(k) ^(T)where:

α is a learning factor,

m1^(k) . . . mn^(k) are n respective additional profile setpoint signalsin a k^(th) iteration,

PS^(k) is a map between the feedforward signal and the output positionsignal in a k^(th) iteration, and ^(T) denotes a transposition.

Next, the convergence of the iteration is considered in more detail.

Given a set of measured impulse responses and a design of the learninggain L^(k), convergence of the iteration procedure can be easilyinvestigated by checking the eigen values of a 4 by 4 matrix. Considerthe update relationship (1). Let the map between the feedforward signaland the output for the real system be given by PS*^(k). Then, (1) can berewritten as follows:

$\begin{matrix}\begin{matrix}{\begin{bmatrix}{kfs}^{k + 1} \\{kfj}^{k + 1} \\{kfa}^{k + 1} \\{kfv}^{k + 1}\end{bmatrix} = {\begin{bmatrix}{kfs}^{k} \\{kfj}^{k} \\{kfa}^{k} \\{kfv}^{k}\end{bmatrix} + {L^{k}\left( {p^{k} - {{{PS}^{*k}\begin{bmatrix}s^{k} & j^{k} & a^{k} & v^{k}\end{bmatrix}}\begin{bmatrix}{kfs}^{k} \\{kfj}^{k} \\{kfa}^{k} \\{kfv}^{k}\end{bmatrix}}} \right)}}} \\{= {{\left\lbrack {I - {L^{k}{{PS}^{*k}\begin{bmatrix}s^{k} & j^{k} & a^{k} & v^{k}\end{bmatrix}}}} \right\rbrack\begin{bmatrix}{kfs}^{k} \\{kfj}^{k} \\{kfa}^{k} \\{kfv}^{k}\end{bmatrix}} + {L^{k}p^{k}}}}\end{matrix} & (5)\end{matrix}$

Equation (5) presents a linear, dynamic system with states kfs^(k),kfj^(k), kfa^(k), and kfv^(k). From linear systems theory, it is wellknown that this system is stable, and hence will converge to a fixedsolution, if the eigen values of the state transition matrix[I−L^(k)PS*^(k)[s^(k)j^(k)a^(k)v^(k)]] are inside the unit circle.

If PS*^(k)=PS^(k) (which is the case if the impulse response of the realsystem is equal to the one that has been used in the design of L^(k)),then from substituting (4) in (5), we find that:

$\begin{matrix}{\begin{bmatrix}{kfs}^{k + 1} \\{kfj}^{k + 1} \\{kfa}^{k + 1} \\{kfv}^{k + 1}\end{bmatrix} = {{\left\lbrack {I - {\alpha\; I}} \right\rbrack\begin{bmatrix}{kfs}^{k} \\{kfj}^{k} \\{kfa}^{k} \\{kfv}^{k}\end{bmatrix}} + {L^{k}p^{k}}}} & (6)\end{matrix}$such that the eigen values of the state transition matrix are equal toα. Convergence is then guaranteed if 0<α<2.

In the most accurate position control systems, a delay compensation maybe present which matches the timing of the feedforward signal with thedelay in the position measurement. This delay can have subsample values.A motion system in accordance with an embodiment of the invention mayeasily incorporate the self-tuning of or adaptation to unknownmeasurement delays. The basic idea is to model the feedforwardcontroller as a linear combination of the snap, jerk, acceleration andvelocity signals and time-shifted versions of those.

An example of the use of delay compensation for an acceleration signalis given in FIG. 3. In this example, the acceleration signal is ‘shiftedback over half a sample’. The original acceleration signal is denoteda₀. a₁ is the same acceleration signal, however shifted back over 1sample. The 0.5 sample shifted acceleration signal is constructed byadding half the original acceleration signal a₀ (second plot of FIG. 3)to half the shifted back acceleration signal a₁ (first plot of FIG. 3).The resulting acceleration signal is shown in the third plot of FIG. 3.

If the sub-sample delay correction is implemented to all four componentsin a typical feedforward signal, then the feedforward signal can beconstructed as follows:

$\begin{matrix}{{FF} = {\begin{bmatrix}s_{0}^{k} & s_{1}^{k} & j_{0}^{k} & j_{1}^{k} & a_{0}^{k} & a_{1}^{k} & v_{0}^{k} & v_{1}^{k}\end{bmatrix}\begin{bmatrix}{kfs}_{0}^{k} \\{kfs}_{1}^{k} \\{kfj}_{0}^{k} \\{kfj}_{1}^{k} \\{kfa}_{0}^{k} \\{kfa}_{1}^{k} \\{kfv}_{0}^{k} \\{kfv}_{1}^{k}\end{bmatrix}}} & (7)\end{matrix}$

In the example shown in FIG. 3, both kfa₀ ^(k) and kfa₁ ^(k) are equalto 0.5.

The eight coefficients of the feedforward controller in (7) can beupdated using an update relationship similar to (1). The derivation ofthis update relationship is omitted. After convergence, an estimate ofthe compensated delay can be obtained as follows:

$\begin{matrix}{\tau = \frac{{kfa}_{1}^{k}}{{kfa}_{0}^{k} + {kfa}_{1}^{k}}} & (8)\end{matrix}$

The effectiveness of the automated delay compensation tuning isillustrated by the following example.

As an example, the application of this technique to the feedforwardtuning for a high precision motion system with actuator flexibility isconsidered. The frequency response function of the motion system isplotted in FIG. 4. To illustrate the robustness of the method, thedesign of the learning gain L is based on a very simple, rigid bodymodel of the motion system, of which the frequency response is shown inFIG. 4 also.

The feedforward controller to be designed consists of a combination ofacceleration and snap feedforward. The measurement delay isapproximately 1 quarter of a sample (the sample time being 1/4000 s))and has to be estimated also. The setpoint snap, acceleration andposition are plotted in FIG. 5. Here, the invention is applied as atuning procedure, with a fixed update gain L.

First, for simplicity, the measurement delay is neglected. In this case,the feedforward controller is given as follows:

${FF}^{k} = {\begin{bmatrix}s^{k} & a^{k}\end{bmatrix}\begin{bmatrix}{kfs}^{k} \\{kfa}^{k}\end{bmatrix}}$

kfs^(k) and kfa^(k) are updated by means of update gain L which isdesigned using (5) with a learning factor α=0.8. The tuning process isinitiated using kfs⁰=0 and kfa⁰=0. Ten iterations are executed. Theresulting errors after 10 iterations are plotted in FIG. 6. The errorwhich results if the snap feedforward is omitted, is also plotted(dotted line). The evolution of kfs^(k) and kfa^(k) over the teniterations is plotted in FIG. 7. Apparently, the feedforward parametersare optimally tuned, despite the mismatch between the motion systemdynamics and the very simple model underlying the design of L.

If the measurement delay has to be taken into account, a feedforwardcontroller can be used as follows:

${FF}^{k} = \left\lbrack \begin{matrix}s_{0}^{k} & s_{1}^{k} & a_{0}^{k} & {\left. a_{1}^{k} \right\rbrack\begin{bmatrix}{kfs}_{0}^{k} \\{kfs}_{1}^{k} \\{kfa}_{0}^{k} \\{kfa}_{1}^{k}\end{bmatrix}}\end{matrix} \right.$where s₀ and a₀ are the original snap and acceleration profile,respectively, and where s₁ and a₁ are the snap and acceleration profileshifted back one sample in time. The resulting error is shown in FIG. 8.To demonstrate the effect of the delay compensation tuning, theconverged error without delay compensation tuning is included in FIG. 8as well (dotted line).

The corresponding evolution of the feedforward coefficients kfa₀ ^(k),kfa₁ ^(k), kfs₀ ^(k), and kfs₁ ^(k) is shown in FIG. 9. The effect ofthe measurement delay is now properly accounted for by the values ofkfa₀ ^(k) and kfa₁ ^(k). Note that after convergence kfa₀ ^(k) is threetimes larger than kfa₁ ^(k). This complies, according to equation (8),with a delay compensation of one quarter of a sample, which is indeedequal to the measurement delay.

The real-time computational demand of the method according to theinvention is very limited. At most 4N^(k) multiplications and 4N^(k)summations have to be performed each trial for a complete update of thefeedforward coefficients. These computations can be spread out overN^(k) samples. If the computations are implemented recursively, forevery sample only 4 multiplications and 4 summations have to be carriedout.

Embodiments of the invention enable the automation of feedforward tuningfor motion control systems. The method can easily be implemented incombination with existing motion control software and eliminates themanual tuning step.

The invention can be applied in lithography stages, PCB assembly robots,the seek control systems of optical drives, hard disk drives, and manyother motion control systems which have to perform setpoint tracking.

Although specific reference may be made in this text to the use oflithographic apparatus in the manufacture of ICs, it should beunderstood that the lithographic apparatus described herein may haveother applications, such as the manufacture of integrated opticalsystems, guidance and detection patterns for magnetic domain memories,flat-panel displays, liquid-crystal displays (LCDs), thin-film magneticheads, etc. The skilled artisan will appreciate that, in the context ofsuch alternative applications, any use of the terms “wafer” or “die”herein may be considered as synonymous with the more general terms“substrate” or “target portion”, respectively. The substrate referred toherein may be processed, before or after exposure, in for example atrack (a tool that typically applies a layer of resist to a substrateand develops the exposed resist), a metrology tool and/or an inspectiontool. Where applicable, the disclosure herein may be applied to such andother substrate processing tools. Further, the substrate may beprocessed more than once, for example in order to create a multi-layerIC, so that the term substrate used herein may also refer to a substratethat already contains multiple processed layers.

Although specific reference may have been made above to the use ofembodiments of the invention in the context of optical lithography, itwill be appreciated that the invention may be used in otherapplications, for example imprint lithography, and where the contextallows, is not limited to optical lithography. In imprint lithography atopography in a patterning device defines the pattern created on asubstrate. The topography of the patterning device may be pressed into alayer of resist supplied to the substrate whereupon the resist is curedby applying electromagnetic radiation, heat, pressure or a combinationthereof. The patterning device is moved out of the resist leaving apattern in it after the resist is cured.

The terms “radiation” and “beam” used herein encompass all types ofelectromagnetic radiation, including ultraviolet (UV) radiation (e.g.having a wavelength of or about 365, 248, 193, 157 or 126 nm) andextreme ultra-violet (EUV) radiation (e.g. having a wavelength in therange of 5–20 nm), as well as particle beams, such as ion beams orelectron beams.

The term “lens”, where the context allows, may refer to any one orcombination of various types of optical components, includingrefractive, reflective, magnetic, electromagnetic and electrostaticoptical components.

While specific embodiments of the invention have been described above,it will be appreciated that the invention may be practiced otherwisethan as described. For example, the invention may take the form of acomputer program containing one or more sequences of machine-readableinstructions describing a method as disclosed above, or a data storagemedium (e.g. semiconductor memory, magnetic or optical disk) having sucha computer program stored therein.

The descriptions above are intended to be illustrative, not limiting.Thus, it will be apparent to one skilled in the art that modificationsmay be made to the invention as described without departing from thescope of the claims set out below.

1. A motion control system, comprising: a motion profile setpointgenerator configured to generate a set of profile setpoint signalsincluding a position profile setpoint signal and additional profilesetpoint signals, and a feedforward controller configured to generate afeedforward signal by summing the additional profile setpoint signals,said additional profile setpoint signals being multiplied by arespective coefficient; wherein the motion control system is configuredto: (a) select an initial setting of the respective coefficients; (b)cause the motion profile setpoint generator to generate a set of profilesetpoint signals so as to execute a motion profile; (c) measure an errorbetween the position profile and an output position signal; and (d)update the respective coefficients by adding a product of the error anda respective learning gain thereto.
 2. The motion control system ofclaim 1, wherein said additional profile setpoint signals comprise firstand higher order derivatives of the position profile setpoint signal. 3.The motion control system of claim 1, wherein the motion control systemis configured to perform a sequence (b)-(c)-(d) k times, where k is apositive integer.
 4. The motion control system of claim 1, wherein therespective coefficients are updated according to the formula:$\begin{bmatrix}{k\; 1^{k + 1}} \\{k\; 2^{k + 1}} \\\vdots \\{kn}^{k + 1}\end{bmatrix} = {\begin{bmatrix}{k\; 1^{k}} \\{k\; 2^{k}} \\\vdots \\{kn}^{k}\end{bmatrix} + {L^{k}e^{k}}}$ where: k1^(k) . . . kn^(k) are respectivecoefficients, in a k^(th) iteration, of n setpoint profile signals,k1^(k+1) . . . kn^(k+1) are respective coefficients, in a (k+1)^(th)iteration, of n setpoint profile signals, L^(k) ε

^(n×N) ^(k) is the learning gain in a k^(th) iteration, N^(k) is anumber of samples in the motion profile used in a k^(th) iteration, ande^(k) is the error in a k^(th) iteration.
 5. The motion control systemof claim 4, wherein the learning gain L^(k) is defined as:L ^(k)=α([m1^(k) m2^(k) . . . mn ^(k)]^(T) PS ^(k) ^(T) PS ^(k) [m1^(k)m2^(k) . . . mn ^(k)])⁻¹ [m1^(k) m2^(k) . . . mn ^(k)]^(T) PS ^(k) ^(T)where: α is a learning factor, m1^(k) . . . mn^(k) are n respectiveadditional profile setpoint signals in a k^(th) iteration, PS^(k) is amap between the feedforward signal and the output position signal in ak^(th) iteration, and ^(T) denotes a transposition.
 6. The motioncontrol system of claim 5, wherein the additional profile setpointsignals are selected from a group of signals consisting of snap, jerk,acceleration and velocity.
 7. The motion control system of claim 5,wherein 0<α<2.
 8. The motion control system of claim 1, wherein therespective coefficients are updated so as to minimize the sum of squaresof the errors.
 9. The motion control system of claim 1, wherein a firstof the additional profile setpoint signals is shifted in time.
 10. Themotion control system of claim 9, wherein a duplicate of an originalsignal of the additional profile setpoint signals is generated, saidduplicate being shifted in time relative to said original signal, andwherein a time-shifted additional profile setpoint signal is added tothe set of profile setpoint signals.
 11. The motion control system ofclaim 1, wherein the motion profile setpoint generator is adapted togenerate a position profile setpoint signal, a velocity profile setpointsignal, an acceleration profile setpoint signal, a jerk profile setpointsignal, and a snap profile setpoint signal.
 12. A method for feedforwardmotion control, comprising: generating a set of profile setpoint signalsincluding a position profile setpoint signal and additional profilesetpoint signals; generating a feedforward signal by summing theadditional profile setpoint signals, said additional profile setpointsignals being multiplied by a respective coefficient, and determiningthe respective coefficients by: (a) selecting an initial setting of therespective coefficients; (b) generating a set of profile setpointsignals so as to execute a motion profile; (c) measuring an errorbetween the position profile and an output position signal; (d) updatingthe respective coefficients by adding a product of the error and arespective learning gain thereto.
 13. The method of claim 12, whereinsaid additional profile setpoint signals comprise first and higher orderderivatives of the position profile setpoint signal.
 14. The method ofclaim 12, further comprising performing the generating of a set ofprofile setpoint signals, the measuring and the updating k times,wherein k is a positive integer.
 15. The method of claim 12, wherein therespective coefficients are updated according to the formula:$\begin{bmatrix}{k\; 1^{k + 1}} \\{k\; 2^{k + 1}} \\\vdots \\{kn}^{k + 1}\end{bmatrix} = {\begin{bmatrix}{k\; 1^{k}} \\{k\; 2^{k}} \\\vdots \\{kn}^{k}\end{bmatrix} + {L^{k}e^{k}}}$ where: k1^(k) . . . kn^(k) are respectivecoefficients, in a k^(th) iteration, of n setpoint profile signals,k1^(k+1) . . . kn^(k+1) are respective coefficients, in a (k+1)^(th)iteration, of n setpoint profile signals, L^(k) ε

^(n×N) ^(k) is the learning gain in a k^(th) iteration, N^(k) is anumber of samples in the motion profile used in a k^(th) iteration, ande^(k) is the error in a k^(th) iteration.
 16. The method of claim 15,wherein the learning gain L^(k) is defined as:L ^(k)=α([m1^(k) m2^(k) . . . mn ^(k)]^(T) PS ^(k) ^(T) PS ^(k) [m1^(k)m2^(k) . . . mn ^(k)])⁻¹ [m1^(k) m2^(k) . . . mn ^(k)]^(T) PS ^(k) ^(T)where: α is a learning factor, m1^(k) . . . mn^(k) are n respectiveadditional profile setpoint signals in a k^(th) iteration, PS^(k) is amap between the feedforward signal and the output position signal in ak^(th) iteration, and ^(T) denotes a transposition.
 17. The method ofclaim 16, wherein the additional profile setpoint signals are selectedfrom a group of signals consisting of snap, jerk, acceleration andvelocity.
 18. The method of control system of claim 16, wherein 0<α<2.19. A lithographic apparatus arranged to transfer a pattern from apatterning device onto a substrate, the lithographic apparatuscomprising: a motion control system configured to control a positioneradapted to position the patterning device or the substrate, the motioncontrol system comprising: a motion profile setpoint generatorconfigured to generate a set of profile setpoint signals including aposition profile setpoint signal and additional profile setpointsignals, and a feedforward controller configured to generate afeedforward signal by summing the additional profile setpoint signals,said additional profile setpoint signals being multiplied by arespective coefficient; wherein the motion control system is configuredto: (a) select an initial setting of the respective coefficients; (b)cause the motion profile setpoint generator to generate a set of profilesetpoint signals so as to execute a motion profile; (c) measure an errorbetween the position profile and an output position signal; and (d)update the respective coefficients by adding a product of the error anda respective learning gain thereto.